Exploring the Mystery of Prime Numbers

[philiprdutton]

I posted the following on my blog (http://fooledbyprimes.blogspot.com/2007/07/silly-primes.html)



Not until recently has the whole prime number "culture" become a distraction to me. While a child the primes never really caught my attention. Even in college there was not much drawing me to the subject beyond the occasional newspaper headline proclaiming the exuberance of the mathematics community as some rather skinny, unkempt math geek held a new largest prime in high esteem.

One of the things that bothered me about primes is how messy they are. From the perspective of where they are on the number line one can't help but get the feeling that any equation related to their distribution is going to be ugly. Maybe I am a sucker for simplicity- just call it an eye for elegance!

Taking a look at the math culture's definition of a prime we find something like: "..a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself." Oh how boring! Of course the mathematicians tell us that primes build all the other numbers. Digging around one will find this formal statement called the fundamental theorem of arithmetic. It says, "every natural number greater than 1 can be written as a unique product of prime numbers." It appears to be very, very important to mathematics- afterall, it is the fundamental theorem of arithmetic!

I must admit I didn't investigate the prime number sequence at all other than taking a quick peek at the first 100 primes. Instead, I became intensely focused on the two related definitions given above. Take a look at the words in the definition and convince yourself which words convey the most "action"- the meat of the definitions so to speak. I came up with "natural number divisors" and "unique product." Now, I must say right away that I failed calculus II so I do not profess to be a brilliant mathematician (don't worry, I took the class again with a different professor and got an passing grade). There is one thing that I do know about math and it is this: multiplication is just repeated addition.

So, I wondered what would happen if the math culture rewrote the fundamental theorem of arithmetic without using the word "product." Wouldn't that be cool- a simplified version of the definition! Maybe... just maybe... we might find some new way to think about prime numbers and make some progress on the stubborn topic.

Personally, I believe that a number which is "prime" is just highlighing a side effect of short-cut addition. We have to have short-cuts otherwise we humans would count to each other when we simply wanted to say "I'll pay you 25 copper coins to feed my camels." Think about the axioms of arithmetic. List them on paper and then erase the ones related to multiplication and division. Now, tell me what a prime number is! I feel that we have been duped by the math community at large because they told us for so long that primes are super important- even godly. I challenge everyone to go back to the basics for the sake of progress! (I know you're just as tired of the centuries-old unsolved prime number mysteries)

What I am saying is that the prime numbers are not mystical. What is mystical is the relationship between the algorithmic process of counting and the notion of short-cuts (multiplication). Are the two different? Yes. Short-cuts require some sort of memory. The memory is in the form of additional "wiring"... like defining new kinds of number systems. Think about it: the Egyptians, Babylonians, Greeks, Hebrews, Hindus, they all count the same. But their short cut methods are what are different. Counting is simple, just repeat after me: "da, da, da, da, da, da, da..." Short-cutting and communicating about where the counting stops is a completely different ballgame and it is what produces the "mysterious" properties that we perceive in the primes.

I would be interested in literature about the primes from the perspective above.

Thanks,

Philip R. Dutton
Columbia, SC, USA
http://fooledbyprimes.blogspot.com/
http://forum.wolframscience.com/member.php?s=&action=getinfo&find=lastposter&forumid=4

 

[HallsofIvy]

I'm sorry: "the fundamental theorem of arithmetic without using the word 'product.'"? I can't imagine how it could be stated more easily! In general, multiplication is NOT a "shortcut" for addition. Thinking it is misses the whole point.

 

[ice109]

how old are you? your writing is worse than high school quality and i don't see your point at all. seems like someone just trying to use a lot of big words.

 

[CRGreathouse]

For what it's worth I see multiplication as the core relation, with addition as a more complex and loess natural system that makes numbers more complex. For me, the primes are quite literally the atom of the natural numbers.

So, I wondered what would happen if the math culture rewrote the fundamental theorem of arithmetic without using the word "product." Wouldn't that be cool- a simplified version of the definition! Maybe... just maybe... we might find some new way to think about prime numbers and make some progress on the stubborn topic.

The most natural such translation that comes to my mind would be using logs. Define lP = {log 2, log 3, log 5, ...}. Now the log of each positive integer can be uniquely represented as a linear combination of values from this set, up to the order of summands. Of course I hardly think logarithms are more natural than products.

Perhaps there is a version of the fundamental theorem using just gcds and its like?

 

[CRGreathouse]

In response to your third blog post's challenge: "Try to define a prime number without using the word 'product' nor the word 'multiplication.'"

A prime number is a number with a nonzero residue modulo all numbers 1 < k < n.

 

[philiprdutton]

ah a breath of fresh air

In response to your third blog post's challenge: "Try to define a prime number without using the word 'product' nor the word 'multiplication.'"

A prime number is a number with a nonzero residue modulo all numbers 1 < k < n.

Interesting! Now we are getting somewhere. I have an idea: Someone should find all the different ways to define "prime". Maybe there would a list of around 10 different fundamental statements depending on your axiomatic system of choice. Surely the list would be beneficial to people like me who are trying to understand the subject but who, clearly can not write well.

 

[CRGreathouse]

Interesting! Now we are getting somewhere. I have an idea: Someone show find all the different ways to define prime number. Maybe there would a list of 10 different fundamental statements depending on your axiomatic system of choice. Surely the list would be beneficial to people like me who are trying to understand the subject but who, clearly can not write well.

Have you taken abstract algebra? You might be interested in the generalization of "prime" and "irreducible", as well as the study of systems where they don't coincide.

 

[philiprdutton]

OFF TOPIC: Writing

how old are you? your writing is worse than high school quality and i don't see your point at all. seems like someone just trying to use a lot of big words.

Thanks for pointing out that my writing is worse than high school quality. I believe you forgot to use "caps" where appropriate. Also, you seem to be using a fragmented sentence. I would tell you how old I am but I prefer to use base 2. If I type out my age in base 2 using a character string of "1"'s and "0"'s then you would probably assume the zero position is on the far right side when in reality, there is nothing preventing me from positioning my zero marker on the far left side. So I will not post my age.

 

[philiprdutton]

prime

Have you taken abstract algebra? You might be interested in the generalization of "prime" and "irreducible", as well as the study of systems where they don't coincide.

Thanks for the tip. I am just a plebeian when compared to a math guru like yourself. Actually, I am just interested in the problem of prime properties and how they relate stated axioms (in whatever system you are using). Consider the Peano axioms. What would happen if you did not define the successor function? Would any given natural number which was prime still be prime if you remove the successor function?

The funny thing about the Peano axioms is that most of them start out with "If b is a natural number..." Well, basically Peano states in his assumptions that you are given all the natural numbers. So, all the natural numbers that happen to be in the position of primes are there too. But if you stop writing down axioms before you define the successor function, then you can not have the notional of primality.

I am having trouble explaining all this. Basically I can create a system for counting with no fluffy extra axioms related to operations. Heck, let me just count to the 100th prime number: "da,da,da,da,da,da,da,da,da,...,da,da,da" There, you see that last "da"? That is in the same position on the number line as the 100th prime number as defined already. However, in my "da, da,da" counting system, I can not tell you what it means to be "prime."

 

[CRGreathouse]

Thanks for pointing out that my writing is worse than high school quality. I believe you forgot to use "caps" where appropriate. Also, you seem to be using a fragmented sentence. I would tell you how old I am but I prefer to use base 2. If I type out my age in base 2 using a character string of "1"'s and "0"'s then you would probably assume the zero position is on the far right side when in reality, there is nothing preventing me from positioning my zero marker on the far left side. So I will not post my age.

I take it your age isn't a base-2 palindrome, then. Assuming you're less than 100 (left-to-right decimal), that narrows it down to {2, 4, 6, 8, 10, 11, 12, 13, 14, 16, 18, 19, 20, 22, 23, 24, 25, 26, 28, 29, 30, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 66, 67, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84. 86, 87, 88, 89, 90, 91, 92, 94, 95, 96, 97, 98}. We're on to you.

 

[CRGreathouse]

Thanks for the tip. I am just a plebeian when compared to a math guru like yourself.

I'm not nearly a guru like Matt Grime, HallsofIvy, or Hurkyl. I only have a bachelor's degree in math -- though I do try to keep up with recent developments.

If you want to read up on abstract algebra, here are some basic notes from the Web:
http://www.math.niu.edu/~beachy/aaol/contents.html

Consider the Peano axioms. What would happen if you did not define the successor function? Would any given natural number which was prime still be prime if you remove the successor function?

Without the successor function you can't show that there are numbers other than 1. You can't define primes, squares, addition, fractions, or anything much.

I am having trouble explain all this. Basically I can create a system for counting with no fluffy extra axioms related to operations. Heck, let me just count to the 100th prime number: "da,da,da,da,da,da,da,da,da,...,da,da,da" There, you see that last "da"? That is in the same position on the number line as the 100th prime number as defined already. However, in my "da, da,da" counting system, I can not tell you what it means to be "prime."

You'll have to be a lot more specific if you want to make sense out of a system weaker than Peano arithmetic.

 

[philiprdutton]

what came first?

Without the successor function you can't show that there are numbers other than 1. You can't define primes, squares, addition, fractions, or anything much.

But I am confused as to why all the peano axioms start out with "if b is a natural number"... Am I missing something? I read Peano and feel as if he assumes all the natural numbers are set into position on the number line even before he finishes all the axioms. I figured his successor function was just a means of getting around. It gets confusing like the chicken and egg dilemma.

Primes, squares, addition, fractions, etc. all have to do with permitted "operations." But I still believe the natural numbers are still implicitly defined and do sit in place on the number line whether the operational axioms are defined yet or not.

 

[philiprdutton]

binary palindromes

I take it your age isn't a base-2 palindrome, then. Assuming you're less than 100 (left-to-right decimal), that narrows it down to {2, 4, 6, 8, 10, 11, 12, 13, 14, 16, 18, 19, 20, 22, 23, 24, 25, 26, 28, 29, 30, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 66, 67, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84. 86, 87, 88, 89, 90, 91, 92, 94, 95, 96, 97, 98}. We're on to you.

Well, if my age happens to be a base-2 palindrome then I know for sure I am not the age of an even number.

 

[philiprdutton]

"Counting is all too easy. Figuring out how to talk about where you stopped is the hard part." - Philip Ronald Dutton

 

[philiprdutton]

multiplication is core

For what it's worth I see multiplication as the core relation, with addition as a more complex ...

Interesting! I must ponder for some time.

 

[CRGreathouse]

But I am confused as to why all the peano axioms start out with "if b is a natural number"... Am I missing something? I read Peano and feel as if he assumes all the natural numbers are set into position on the number line even before he finishes all the axioms. I figured his successor function was just a means of getting around. It gets confusing like the chicken and egg dilemma.

The Peano axioms say that

• 1 is a natural number
• For all x, Sx is a natural number
• For all x, Sx is not 1
• For all x and y, x = y iff Sx = Sy

plus a number of things not relevant here.

The successor function is the only way to create new numbers in this system. The last property makes each number 1, S(1), S(S(1)), ... different.

There's really no chicken-egg problem -- unless you remove the successor operation. If you do that you'll need to add in a lot of tools to do most anything.

Primes, squares, addition, fractions, etc. all have to do with permitted "operations." But I still believe the natural numbers are still implicitly defined and do sit in place on the number line whether the operational axioms are defined yet or not.

That's a philosophical statement, not a mathematical one. It's called Platonism and is largely out of favor today -- though I consider myself largely a mathematical platonist.

 

[philiprdutton]

what the heck is x?

The Peano axioms say that

• 1 is a natural number
• For all x, Sx is a natural number
• For all x, Sx is not 1
• For all x and y, x = y iff Sx = Sy

plus a number of things not relevant here.

I am now confused about what "x" is. If within the Peano system, there are only natural numbers, then surely x is a natural number.

PS: thanks for chatting thus far!

 

[CRGreathouse]

I am now confused about what "x" is. If within the Peano system, there are only natural numbers, then surely x is a natural number.

I presume you refer to "for all x, Sx is a natural number". That x is a natural number is obvious; that there is a successor to it is not obvious. The statement is essentially that there is a successor to every natural number,

 

[philiprdutton]

* 1 is a natural number
* For all x, Sx is a natural number
* For all x, Sx is not 1
* For all x and y, x = y iff Sx = Sy

Okay. So regarding the obscure number line that sort of exists before Peano touches the paper with his pencil, I imagine what would happen if Peano wrote the following:

* 32654 is a natural number
* For all x, Sx is a natural number
* For all x, Sx is not 32654
* For all x and y, x = y iff Sx = Sy

It would be totally cool with me. But it sort of points out that the number line is still there regardless of whether Peano writes the axioms down or not. Sure I take your point that it is a rather platonistic statement but I can not separate out the platonisticism when talking about this stuff at this level.

In fact, just for fun, I will add a few more:

* 99 is a natural number
* For all x, Sx is a natural number
* For all x, Sx is not 99
* For all x and y, x = y iff Sx = Sy
* 10010001 is a natural number
* For all x, Sx is a natural number
* For all x, Sx is not 10010001
* For all x and y, x = y iff Sx = Sy

* 666 is a natural number
* For all x, Sx is a natural number
* For all x, Sx is not 666
* For all x and y, x = y iff Sx = Sy

(the above I could not resist!)

 

[CRGreathouse]

"1" is just a label, it could be anything. Consider this model of Peano arithmetic:

"1" is a triangle. If x is a polygon, Sx is a polygon with one more side than x; otherwise, Sx is a pink unicorn. x = y iff x and y are polygons and x and y have the same number of sides. For example S(S(1)) (that is, "3") is a pentagon.

This works perfectly well -- all the Peano axioms can be made to hold in this system, even though the underlying objects are not "numbers" in any normal sense of the word. If we went through all the usual definition we would find that x + y would be defined as a polygon with two fewer sides than the total number of sides in x and y.

 

[philiprdutton]

Yes I am in agreement with you about the labels. But, I am not convinced that Peano wrote that stuff without already having been biased by the notion of the number line.

Also, given

* 1 is a natural number
* For all x, Sx is a natural number
* For all x, Sx is not 1
* For all x and y, x = y iff Sx = Sy

I still maintain the early phrase "For all x" is unfortunate. It clearly can be interpreted as saying "for all the x that already exist." This is very much a problem in my opinion. It feels as if the system is riding on the edge of chicken and egg. I am sorry but it will take me some time to absorb that it is not a chicken and egg system (I will take the mathematicians word for it for now).

If, however, the numbers can exist without the successor function definition (example: in terms of counting only and not in terms of talking about where the counting stops) then clearly there is a problem with the definition of Prime. It refers not to the "position" where the counting stops UNTIL the operational axioms are applied.

 

[CRGreathouse]

Yes I am in agreement with you about the labels. But, I am not convinced that Peano wrote that stuff without already having been biased by the notion of the number line.

I'm not entirely sure what you mean. I do think he gave examples of non-numbers as a basis for the Peano axioms, though.

I still maintain the early phrase "For all x" is unfortunate. It clearly can be interpreted as saying "for all the x that already exist." This is very much a problem in my opinion. It feels as if the system is riding on the edge of chicken and egg. I am sorry but it will take me some time to absorb that it is not a chicken and egg system (I will take the mathematicians word for it for now).

Philosophically it may pose a problem, but as formally phrased (and I haven't used the formal phrasings or notations) it is mathematically airtight.

If, however, the numbers can exist without the successor function definition (example: in terms of counting only and not in terms of talking about where the counting stops) then clearly there is a problem with the definition of Prime. It refers not to the "position" where the counting stops UNTIL the operational axioms are applied.

Hmm. I don't know what you mean, but this seems like philosophy again. If you would rephrase this in more detail perhaps I can say something. If it's meant to be mathematical I'll need to know the precise axioms you're using (if not successor) and how you define prime (and how you define everything you use to define prime). When modifying systems precision is very important.

 

[philiprdutton]

Let us assume that there can be an axiomatic system which does not allow you to construct the natural numbers. Assume also that the axiomatic system just states all the would-be numbers in terms of "steps" in a counting process.

Obviously, why do this since the point of the axiomatic system is to be able to build numbers. Don't worry about that for now.

So here is the "foo" axiomatic system:

"foo" axioms:
"foo"
"foo,foo"
"foo,foo,foo"
"foo,foo,foo,foo"
"foo,foo,foo,foo,..."

That was my best attempt at defining a simple basic axiomatic system which just states the steps at each count of a counting process.

Obviously, to talk about each step in terms of a number one needs to have some structure about what it is that you are talking about. Once you add the structure you can finally say that "foo,foo,foo,foo,foo" is really 5. But until you have done so, the "foo,foo,foo,foo,foo" is not a prime even though it falls in the same position on the number line as the 5 from the other axiomatic system.

It is not a formal example but I am trying my best to get to that point. Any ideas how we can formally write an axiomatic system which just lists the "foo" ?? Perhaps we can call it the "counting axioms."

 

[CRGreathouse]

It is not a formal example but I am trying my best to get to that point. Any ideas how we can formally write an axiomatic system which just lists the "foo" ?? Perhaps we can call it the "counting axioms."

I think you just did that -- mathematicians would use the term "axiom schema", that is, each number is its own axiom:

Axiom 1: 1 is a natural number.
Axiom 2: 2 is a natural number.
. . .

OK, so now you have a system where you cannot add or take the successor, but you have the natural numbers. (You can call them "foo, foo" and the like if you wish, but names are just labels to mathematicians so I'll just call them this for now.)

So in the context of this system containing your counting axiom schema, what is the definition of "prime"? Or is your point that you can't even define it here? (Mathematicians would say, informally, that they don't have the 'machinery' they need.)

 

[philiprdutton]

how to define prime?

So in the context of this system containing your counting axiom schema, what is the definition of "prime"? Or is your point that you can't even define it here? (Mathematicians would say, informally, that they don't have the 'machinery' they need.)

Yes. It is my point. In that system there is no way to define prime. Hence, primality is not a feature until more machinery is added.

 

[Hurkyl]

From the point of view of formal logic, having a multiplication operation makes a rather large difference.

If we use first-order logic, and we omit general multiplication, we have the theory of Presburger arithmetic. This theory is known to be logically complete. But if we use Peano arithmetic... or even if we omit the induction axiom and use Robinson arithmetic, then we are working with an incomplete theory.

In other words, if we stick only to addition, every (first-order) proposition we can state about the natural numbers can either be proven or disproven. But if we allow multiplication, then there exist statements that can neither be proven or disproven. (And furthermore, it remains incomplete, even if we adopt finitely more axiom schema)


I don't know much about what happens when you allow higher-order logic.

 

[Hurkyl]

But, I am not convinced that Peano wrote that stuff without already having been biased by the notion of the number line.

Well, of course Peano was thinking about the "number line". His goal (I presume) was to write down a list of axioms that characterized the intuitive notion of "natural number" that he and other mathematicians had.

The modern approach to mathematics prefers to have explicit foundations -- these days, one often defines that anything satisfying the Peano axioms is a "set of natural numbers", or similar. Then, if we turn to metamathematics, we argue that the counting process does, in fact, satisfy the Peano axioms, and so we are justified in saying that when we count, we are using the natural numbers.

 

[CRGreathouse]

If we use first-order logic, and we omit general multiplication, we have the theory of Presburger arithmetic. This theory is known to be logically complete.

Presburger arithmetic may be of interest to philiprdutton because it's just 'on the edge' of being able to define primes. It can't define the concept of being prime in general, let alone prove statements about them, but it can (I think) show that particular numbers are prime:

$$\neg\exists x : x + x = 5$$
$$\neg\exists x : x + x + x = 5$$
$$\neg\exists x : x + x + x + x = 5$$

 

[philiprdutton]

mapping

I think you just did that -- mathematicians would use the term "axiom schema", that is, each number is its own axiom:

Axiom 1: 1 is a natural number.
Axiom 2: 2 is a natural number.
. . .

OK, so now you have a system where you cannot add or take the successor, but you have the natural numbers. ...

Finally I had one last important thought. Given the system described where you have the natural numbers but you can not add or take the successor, we should be able to map the system to the system where by you can build the natural numbers. If such a mapping is "formalized" then the problem appears. On the one hand you have a system where "prime" is not defined and on the other hand you have a system where "prime" can be defined. They are mapped to each other and so now is there a paradox?

I am thinking about the utility of mapping similar to what is used by Godel in his famous proof.

Again for clarity: we defined a "counting" style, infinite statement axiomatic system which you have no notion of multiplication nor successor function (as in the above posts). We have another system like Peano. Both systems produce something that lies on the same place on the number line. We use mapping to link the two systems through the "number line." Now, despite the mapping (if it is possible), you can not impose the notion of prime on the simpler system. Hence, the notion of "prime" is directly related to the mechanisms of addition/multiplication or other operations... NOT the actually position on the number line thing.

 

[philiprdutton]

"The idea of a prime number is loads of fun for the guy with all the numbers AND the bag of tools with which he can do things to those numbers. The guy with only all the numbers is simply bored out of his mind." - Philip Ronald Dutton

(sorry! I am exploiting the utility of writing hoping it will sharpen my understanding of all this)

 

[CRGreathouse]

we defined a "counting" style, infinite statement axiomatic system which you have no notion of multiplication nor successor function (as in the above posts). We have another system like Peano. Both systems produce something that lies on the same place on the number line. We use mapping to link the two systems through the "number line." Now, despite the mapping (if it is possible), you can not impose the notion of prime on the simpler system. Hence, the notion of "prime" is directly related to the mechanisms of addition/multiplication or other operations... NOT the actually position on the number line thing.

But of course. I can also set up a linking from the "counting" (on the left) to Peano Arithmetic (on the right) like so:

1 <--> 3
2 <--> 2
3 <--> 1
4 <--> 6
5 <--> 5
6 <--> 4
7 <--> 9
. . .

The mapping is perfectly reasonable, and all properties (i.e. none) that held in the counting system still hold in Peano arithmetic. The counting numbers that are prime in PA, though, are 1, 2, 5, and so on -- not at all the same.

 

[philiprdutton]

branches

But of course. I can also set up a linking from the "counting" (on the left) to Peano Arithmetic (on the right) like so:

1 <--> 3
2 <--> 2
3 <--> 1
4 <--> 6
5 <--> 5
6 <--> 4
7 <--> 9
. . .

The mapping is perfectly reasonable, and all properties (i.e. none) that held in the counting system still hold in Peano arithmetic. The counting numbers that are prime in PA, though, are 1, 2, 5, and so on -- not at all the same.

May I ask how you start with 3 and then get 2... 1,6,5,4,9?

Also, is the counting system single branch (of statements) as opposed to a multi-branch PA tree of types of statements? If each axiom in PA can produce a certain amount of statements then that set of statements is what I am informally calling a branch. Since the counting system only has one way to make statements it is single branch.

 

[dodo]

If you want to play "da da da da" for a while, stress one "da" of every N, as in "da da DA da da DA ..."; if you put them all together,

2 da DA da DA da DA da DA da DA da DA da ...
3 da da DA da da DA da da DA da da DA da da DA ...
4 da da da DA da da da DA da da da DA da da da DA ...
5 da da da da DA da da da da DA da da da da DA da da da da DA ...
6 da da da da da DA da da da da da DA da da da da da DA ...
7 da da da da da da DA da da da da da da DA da da da da da da DA ...

a prime number is one where the first stressed DA's won't coincide with any DA for all smaller numbers.

(Which of course is a re-edition of the [/PLAIN]
Sieve of Eratosthenes.)

 

[CRGreathouse]

May I ask how you start with 3 and then get 2... 1,6,5,4,9?

Yes, but that would be beside the point. They have no properties, so there's nothing making the counting "2" more or less like the Peano "2" than the Peano "7". I can put them in any order I want -- and in fact I could associate them with only the Peano primes, or only the Peano composites, or only the Peano powers of 2 that are squares.

Also, is the counting system single branch (of statements) as opposed to a multi-branch PA tree of types of statements? If each axiom in PA can produce a certain amount of statements then that set of statements is what I am informally calling a branch. Since the counting system only has one way to make statements it is single branch.

Terminology. Remember that both Peano arithmetic and your counting system have an infinite number of axioms -- you have one axiom schema, which actually has omega members (one for each natural number). So yes, each of your axioms has only one statement it can make, but you can make an infinite number of statements.

That aside, I'm still not sure I quite follow. What is the motivation behind the branch terminology?

 

[Hurkyl]

Incidentally, as far as formal logic is concerned, the axiomatic method is merely a convenient way for presenting formal theories. There is no inherent quality of a statement that determines whether or not it is an axiom -- it's simply an artifact of the way the formal theory is presented.